Loop-Type Wave-Generation Method for Generating Wind Waves under Long-Fetch Conditions
1. Introduction
It is very important to predict the momentum, heat, and mass transfer across the air–sea interface during tropical cyclones, as the transfer mechanism has significant influence on the accuracy of the predictions of future climate change, the intensity of the tropical cyclones, and other meteorological phenomena. In particular, the air–sea momentum, heat, and mass transfer at high wind speeds is significantly affected by wave breaking, and it is therefore essential to understand the formation and shape of wind waves at high wind speeds (Takagaki et al. 2016a,b).
Therefore, the purpose of this study is to develop an original technique for generating wind waves with long fetches in a laboratory for discussing the air–sea momentum and scalar transfer in the wide wind speed region, including extremely high wind speeds. The laboratory experiments in such long-fetch conditions would be helpful for modeling small-scale air–sea coupling.
2. Experiment
a. Equipment and measurement methods
A high-speed wind-wave tank (HSWWT) (cf. Takagaki et al. 2012; Iwano et al. 2013; Krall and Jähne 2014) with a glass test section 15.0 m long, 0.8 m wide, and 1.6 m high was used in the experiments (see Fig. 1). Wind waves were generated in a water tank, filled with filtered tap water, at U10 = 19.3, 32, and 42 m s−1 with 2.5% variation. Mechanical waves were generated using a programmable piston-type irregular-wave generator constructed using a wave-generating board, a servomotor (Mitsubishi Electric HC-SFS-352), a function generator (NF Corporation circuit WF1973), a wave gauge, a data recorder, and a computer. The wave-generating board was an acrylic plate with a height, width, and thickness of 0.72, 0.78, and 0.02 m, respectively. The center of the board stroke was set to be x = −0.5 m under the entrance slope, and the maximum stroke was 0.4 m.
(a) Schematic diagram of HSWWT and programmable irregular-wave generator, and (b) LTWGM: (top) first wind waves generated by the fan; (middle) second wind waves generated by the wave generator and fan, and the first wind-wave record; and (bottom) third wind waves generated by the wave generator and fan, and the second wind-wave record.
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
Water-level fluctuations were measured using resistance-type wave gauges (Kenek CHT4-HR60BNC). The resistance wire was placed in the water, and the electrical resistance at the instantaneous water level was recorded at 500 Hz for 600 s using a digital recorder (Sony EX-UT10). The energy of the wind waves E was estimated by integrating the spectrum of water-level fluctuations over the frequency f, where the peak frequency of the wind waves fm was defined as the peak of the spectrum. To measure the wavelength LS and the phase velocity CP of significant wind waves, another wave gauge was fixed downstream at Δx = 0.02 m for U10 < 30 m s−1 and at Δx = 0.19 m for U10 
The measurements of velocity and water-level fluctuations were carried out at a distance of x = 5.5, 6.5, 8.5, or 11.5 m from the edge (x = 0 m) of an entrance slope plate (Fig. 1). Wave absorbers were positioned at the inlet and outlet of the test section to prevent the reflection of surface waves.
b. LTWGM
Spectral models proposed for normal wind speeds (e.g., Phillips 1958; Pierson and Moskowitz 1964; Hasselmann et al. 1973; Toba 1973) are not appropriate for the generation of wind waves at extremely high wind speeds, as the wind-wave properties thereof are not well known. Therefore, a new method, referred to as the loop-type wave-generation method (LTWGM), was employed to generate waves under long-fetch conditions in HSWWT (see Fig. 1b). The LTWGM was used to extend the actual fetch incrementally. First, wind waves were generated using a fan without the mechanical irregular-wave generator, and the water-level fluctuation was measured at Xloop = 11.5 m (see Figs. 1b, top; Table 1). The measured spectrum at Xinitial = 11.5 m was used as input (see Table 1) for the irregular-wave generator positioned at the inlet of the test section (see Fig. 1b, middle). Then, irregular wind waves were generated mechanically and forced with the fan starting at a fetch of 11.5 m and measured at 17 m (F = Xinitial + Xmeasure = 11.5 + 5.5 = 17 m, where Xmeasure means the measurement location of waves for wave analysis; see Table 1). This iterative procedure was repeated to generate wind waves at different values of F (Fig. 1b, bottom). Wind waves were generated at F = 6.5, 17, 25.5, 34, and 42.5 m using LTWGM 0, 1, 2, 3, and 4 times, respectively. The measurement locations are listed in Table 1. In this study, the conditions with F = 6.5 m and F > 6.5 m are referred to as short- and long-fetch conditions, respectively, for distinguishing pure wind-driven waves with a short fetch from the wind waves with a long fetch generated by LTWGM. As the maximum LS in HSWWT is approximately 1.22 m, the ratio (D/LS) of the water depth D (=0.62–0.74 m) to the wavelength is 0.5. This satisfies a strict criterion for deep-sea waves (Lamb 1932). This also implies that the wind waves generated in HSWWT by LTWGM could be regarded as deep-sea waves under all laboratory conditions used in this study.
Measurement conditions: U10: wind speed at a height of 10 m above the sea surface, CD: drag coefficient, N: number of loops, F: fetch (=Xinitial + Xmeasure), Xinitial: extended length for LTWGM, Xmeasure: measurement location of waves for wave analysis this time, Xloop: measurement location of waves for LTWGM next time.
c. Wave-generation method
To generate wind waves using LTWGM, it is important that the power spectrum of the waves produced by the wave generator is the same as that measured. The steps for employing LTWGM are as follows:
Relationship between the monochromatic Ei(fi) and Li(fi), and the empirical model for F(fi), as shown in Eqs. (13)–(15).
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
d. Verification experiments
To verify the wind waves generated by LTWGM in HSWWT, we provided additional measurements in another long wind-wave tank [Research Institute for Applied Mechanics, Kyusyu University (RIAM), wind-wave tank (WWT)] with a 54-m-long test section. We measured the mean wind speed at F = 6.5, 20, and 35 m using a pitot tube and an electric differential pressure gauge (Testo 6321), and estimated U10 using the profile method with the log law [Eq. (2)]. The sampling frequency and the sampling time were 1 Hz and 120 s, respectively. In addition, we measured the water-level fluctuation using custom capacitance-type wave gauges with Δx = 0.15 m for F = 6.5 m and at Δx = 0.3 m for F = 20, 35 m. The preliminarily relationship between the output voltages of the gauges and water levels were calibrated. The sampling frequency and the sampling time were 100 Hz and 360 s, respectively. The energy, peak frequency, wavelength, and phase velocity of significant wind waves were evaluated by spectral and cospectra methods, as with the case in HSWWT (see the details in section 2a). Output voltages of the differential pressure gauge and wave gauges were recorded using a digital recorder (Teac PS2032GP).
3. Results
To verify that the laboratory wind waves generated using LTWGM are similar to those observed in the ocean, we have to investigate the wind-sea spectrum shape, fetch law, dispersion relation, and Toba’s 3/2 power law (Toba 1972), since these laws are observed for pure wind-driven waves in both the ocean and the laboratory. Figure 3 shows the wind-sea spectra at U10 = 19.3, 32.0, and 42.0 m s−1 for several fetch conditions. Here, to clarify the slope at the spectral tails, Figs. 3b,d,f show the spectra multiplied by the fourth power of the frequency against the frequencies normalized by the peak frequency. It is observed that the wind-sea spectra under long-fetch conditions consist of a single peak as a result of significant waves and spectra under short-fetch conditions, and that the peak frequency decreases with increasing the fetch. This is attributed to the development of wind waves owing to the increase in the fetch. In addition, although waves with frequencies higher than 3.5 Hz cannot be produced by the present wave generator, the slope at frequencies higher than fm corresponds to a value of −4 in the frequency range from f/fm = 1.5 to 4 (see Figs. 3b,d,f). This implies that despite the generation of mechanical irregular waves by the wave generator, the wind shear itself produces the condition of local equilibrium between the wind shear and the wind waves generated by LTWGM (Toba 1973). Therefore, it is concluded that LTWGM using the programmable irregular-wave generator can generate the ideal wind waves under long-fetch conditions. Figures 3g,h show the comparisons of spectra of waves generated by LTWGM in HSWWT to those of pure wind-generated waves by a fan in RIAM WWT. It is found that those spectra of the wind waves by LTWGM correspond approximately to those of the pure wind-driven waves, although 1) the primary spectral peak size of pure wind-generated waves by the fan in the long RIAM WWT is larger than that of the wind waves generated by LTWGM in HSWWT at a long fetch of F = 35 m; and 2) the secondary peak at f = 1.2 Hz exists just on the spectrum of the fan in RIAM WWT at a short fetch of F = 6.5 m. The fetch relationship for the size of the wind waves is due to the initial difference as discussed later (Fig. 3g). The second difference in the secondary peak at f = 1.2 Hz at F = 6.5 m is due to the multiple wave reflection in RIAM WWT. Although we could not remove the reflection, the problem is unrelated to the establishment of LTWGM. In addition, although the −4 slope is maintained between f/fm = 1.5 and 4 in Figs. 3b,h, the slopes become more gentle than −4 over f/fm = 4. This may be due to the effects of bound parasitic capillary waves with phase velocities that are close to those of peak waves (e.g., Rozenberg et al. 1999).
Wind-sea spectra Sηη at (a),(b) normal wind speeds (U10 = 19.3 m s−1), (c),(d) normal wind speeds (U10 = 32.0 m s−1), and (e),(f) extremely high wind speeds (U10 = 42.0 m s−1); and (g),(h) comparison of wind waves generated by fan and the wave generator in HSWWT (solid curves) to those by fan in RIAM WWT (dotted curves) at normal wind speeds. Curves, from top to bottom, are for F = 42.5, 34, 25.5, 17, and 6.5 m in (a) and (b); F = 25.5, 17, and 6.5 m in (c) and (d); F = 17 and 6.5 m in (e) and (f). In (b),(d),(f), and (h), the vertical axes show spectra multiplied by the fourth power of the frequency, horizontal axes show frequencies normalized by fm, and dotted lines show horizontal lines. Spectra at high wind speeds [(c)–(f)] are described for f = 5 Hz by removing the impingement effects of droplets and bubbles on water-level measurements (e.g., Takagaki et al. 2016a). In (g) and (h), solid curves, from top to bottom, are for F = 34, 20, and 6.5 m at U10 = 19.3 m s−1, and dotted curves are for F = 35, 20, and 6.5 m at U10 = 21.7 m s−1. Spectra are offset for the clarity in (a)–(h), and solid lines represent a slope of −4 in (a),(c),(e), and (g).
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
Relationship between β and uag−1/2. Values measured in HWWT (solid squares, circles, and triangles) and RIAM WWT (solid diamonds). Shown also are the fetch (colored bars), and the best-fit line by Resio et al. (2004; solid line), their range of values observed (dotted line), and the best-fit line by Romero and Melville (2010; dashed line).
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
Figure 5 shows the vertical distribution of the mean wind speed and the Reynolds stress for F = 6.5, 25.5, and 42.5 m at U10 = 19.3 m s−1. Large waves generated by the wave generator may disturb an inner layer of the airflow field near the water surface by turbulent mixing due to large surface waves. However, in the figure, we can confirm the ideal turbulent boundary airflow that has the free-stream wind region, with the zero Reynolds stress, and the log-law region, with the Reynolds stress decreasing with height z under all the fetch conditions.
Vertical distributions of (a) streamwise mean wind speed U and (b) Reynolds stress 
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
Figure 6 shows that the relationships of the nondimensional fetch 
Relationship between 
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
(a) Dispersion relation, and (b) relationship between T* and H*. In (b), wave height and wave period are normalized using u* and g. Values measured in HWWT (solid squares, circles, and triangles) and RIAM WWT (solid diamonds). In (a) and (b), the dispersion relation and Toba’s 3/2 power law (Toba 1973), respectively, are shown (solid curve). In (b) the 20% errors in Toba’s 3/2 power law (Toba 1973) are shown (dashed curves). The fetch is indicated (colored bars). Laboratory values of Troitskaya et al. (2012), predicted values of Takagaki et al. (2015) and Kurose et al. (2016), and field values of Mitsuyasu and Nakayama (1969) and Johnson et al. (1998) are added in the figure.
Citation: Journal of Atmospheric and Oceanic Technology 34, 10; 10.1175/JTECH-D-17-0043.1
4. Discussion
The abovementioned verifications confirmed ideal wind-driven waves under long-fetch conditions can be generated using LTWGM with the programmable irregular-wave generator. Unfortunately, present long-fetch conditions produced in the laboratory experiments with LTWGM do not correspond perfectly to the field conditions. The differences between the present laboratory conditions and the field conditions are as follows. 1) The initial airflow entering the test section in each loop has a new turbulent boundary layer above the air–water interface under the present laboratory conditions; however, the boundary layer continues to develop under field conditions. 2) The present wave generator with LTWGM in HSWWT with a width of 0.8 m cannot produce the angular wave-sea spectrum, although the actual ocean waves spread to angular directions under field conditions. 3) The slope of the spectral tail is more gentle than −4 over f/fm = 4 (see Figs. 3b,h), although the tail often maintains the −4 slopes under field conditions. These points should be improved before regarding the long-fetch laboratory conditions as ocean conditions, but the present LTWGM would be helpful in laboratory experiments for modeling of small-scale air–sea coupling by means of laboratory experiments.
Acknowledgments
This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (Grants-in-Aid 25249013, 16K18015). We thank H. Muroya and S. Urakawa for their help in conducting experiments at HSWWT, Kyoto University. The authors acknowledge A. Isobe, K. Yufu, M. Ishibashi, and K. Takane for their help in conducting the laboratory measurements at RIAM WWT, Kyushu University.
APPENDIX
Equilibrium Range Constant
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